amos/zbesj.f

   1       SUBROUTINE ZBESJ(ZR, ZI, FNU, KODE, N, CYR, CYI, NZ, IERR)
   2 C***BEGIN PROLOGUE  ZBESJ
   3 C***DATE WRITTEN   830501   (YYMMDD)
   4 C***REVISION DATE  890801   (YYMMDD)
   5 C***CATEGORY NO.  B5K
   6 C***KEYWORDS  J-BESSEL FUNCTION,BESSEL FUNCTION OF COMPLEX ARGUMENT,
   7 C             BESSEL FUNCTION OF FIRST KIND
   8 C***AUTHOR  AMOS, DONALD E., SANDIA NATIONAL LABORATORIES
   9 C***PURPOSE  TO COMPUTE THE J-BESSEL FUNCTION OF A COMPLEX ARGUMENT
  10 C***DESCRIPTION
  11 C
  12 C                      ***A DOUBLE PRECISION ROUTINE***
  13 C         ON KODE=1, CBESJ COMPUTES AN N MEMBER  SEQUENCE OF COMPLEX
  14 C         BESSEL FUNCTIONS CY(I)=J(FNU+I-1,Z) FOR REAL, NONNEGATIVE
  15 C         ORDERS FNU+I-1, I=1,...,N AND COMPLEX Z IN THE CUT PLANE
  16 C         -PI.LT.ARG(Z).LE.PI. ON KODE=2, CBESJ RETURNS THE SCALED
  17 C         FUNCTIONS
  18 C
  19 C         CY(I)=EXP(-ABS(Y))*J(FNU+I-1,Z)   I = 1,...,N , Y=AIMAG(Z)
  20 C
  21 C         WHICH REMOVE THE EXPONENTIAL GROWTH IN BOTH THE UPPER AND
  22 C         LOWER HALF PLANES FOR Z TO INFINITY. DEFINITIONS AND NOTATION
  23 C         ARE FOUND IN THE NBS HANDBOOK OF MATHEMATICAL FUNCTIONS
  24 C         (REF. 1).
  25 C
  26 C         INPUT      ZR,ZI,FNU ARE DOUBLE PRECISION
  27 C           ZR,ZI  - Z=CMPLX(ZR,ZI),  -PI.LT.ARG(Z).LE.PI
  28 C           FNU    - ORDER OF INITIAL J FUNCTION, FNU.GE.0.0D0
  29 C           KODE   - A PARAMETER TO INDICATE THE SCALING OPTION
  30 C                    KODE= 1  RETURNS
  31 C                             CY(I)=J(FNU+I-1,Z), I=1,...,N
  32 C                        = 2  RETURNS
  33 C                             CY(I)=J(FNU+I-1,Z)EXP(-ABS(Y)), I=1,...,N
  34 C           N      - NUMBER OF MEMBERS OF THE SEQUENCE, N.GE.1
  35 C
  36 C         OUTPUT     CYR,CYI ARE DOUBLE PRECISION
  37 C           CYR,CYI- DOUBLE PRECISION VECTORS WHOSE FIRST N COMPONENTS
  38 C                    CONTAIN REAL AND IMAGINARY PARTS FOR THE SEQUENCE
  39 C                    CY(I)=J(FNU+I-1,Z)  OR
  40 C                    CY(I)=J(FNU+I-1,Z)EXP(-ABS(Y))  I=1,...,N
  41 C                    DEPENDING ON KODE, Y=AIMAG(Z).
  42 C           NZ     - NUMBER OF COMPONENTS SET TO ZERO DUE TO UNDERFLOW,
  43 C                    NZ= 0   , NORMAL RETURN
  44 C                    NZ.GT.0 , LAST NZ COMPONENTS OF CY SET  ZERO DUE
  45 C                              TO UNDERFLOW, CY(I)=CMPLX(0.0D0,0.0D0),
  46 C                              I = N-NZ+1,...,N
  47 C           IERR   - ERROR FLAG
  48 C                    IERR=0, NORMAL RETURN - COMPUTATION COMPLETED
  49 C                    IERR=1, INPUT ERROR   - NO COMPUTATION
  50 C                    IERR=2, OVERFLOW      - NO COMPUTATION, AIMAG(Z)
  51 C                            TOO LARGE ON KODE=1
  52 C                    IERR=3, CABS(Z) OR FNU+N-1 LARGE - COMPUTATION DONE
  53 C                            BUT LOSSES OF SIGNIFCANCE BY ARGUMENT
  54 C                            REDUCTION PRODUCE LESS THAN HALF OF MACHINE
  55 C                            ACCURACY
  56 C                    IERR=4, CABS(Z) OR FNU+N-1 TOO LARGE - NO COMPUTA-
  57 C                            TION BECAUSE OF COMPLETE LOSSES OF SIGNIFI-
  58 C                            CANCE BY ARGUMENT REDUCTION
  59 C                    IERR=5, ERROR              - NO COMPUTATION,
  60 C                            ALGORITHM TERMINATION CONDITION NOT MET
  61 C
  62 C***LONG DESCRIPTION
  63 C
  64 C         THE COMPUTATION IS CARRIED OUT BY THE FORMULA
  65 C
  66 C         J(FNU,Z)=EXP( FNU*PI*I/2)*I(FNU,-I*Z)    AIMAG(Z).GE.0.0
  67 C
  68 C         J(FNU,Z)=EXP(-FNU*PI*I/2)*I(FNU, I*Z)    AIMAG(Z).LT.0.0
  69 C
  70 C         WHERE I**2 = -1 AND I(FNU,Z) IS THE I BESSEL FUNCTION.
  71 C
  72 C         FOR NEGATIVE ORDERS,THE FORMULA
  73 C
  74 C              J(-FNU,Z) = J(FNU,Z)*COS(PI*FNU) - Y(FNU,Z)*SIN(PI*FNU)
  75 C
  76 C         CAN BE USED. HOWEVER,FOR LARGE ORDERS CLOSE TO INTEGERS, THE
  77 C         THE FUNCTION CHANGES RADICALLY. WHEN FNU IS A LARGE POSITIVE
  78 C         INTEGER,THE MAGNITUDE OF J(-FNU,Z)=J(FNU,Z)*COS(PI*FNU) IS A
  79 C         LARGE NEGATIVE POWER OF TEN. BUT WHEN FNU IS NOT AN INTEGER,
  80 C         Y(FNU,Z) DOMINATES IN MAGNITUDE WITH A LARGE POSITIVE POWER OF
  81 C         TEN AND THE MOST THAT THE SECOND TERM CAN BE REDUCED IS BY
  82 C         UNIT ROUNDOFF FROM THE COEFFICIENT. THUS, WIDE CHANGES CAN
  83 C         OCCUR WITHIN UNIT ROUNDOFF OF A LARGE INTEGER FOR FNU. HERE,
  84 C         LARGE MEANS FNU.GT.CABS(Z).
  85 C
  86 C         IN MOST COMPLEX VARIABLE COMPUTATION, ONE MUST EVALUATE ELE-
  87 C         MENTARY FUNCTIONS. WHEN THE MAGNITUDE OF Z OR FNU+N-1 IS
  88 C         LARGE, LOSSES OF SIGNIFICANCE BY ARGUMENT REDUCTION OCCUR.
  89 C         CONSEQUENTLY, IF EITHER ONE EXCEEDS U1=SQRT(0.5/UR), THEN
  90 C         LOSSES EXCEEDING HALF PRECISION ARE LIKELY AND AN ERROR FLAG
  91 C         IERR=3 IS TRIGGERED WHERE UR=DMAX1(D1MACH(4),1.0D-18) IS
  92 C         DOUBLE PRECISION UNIT ROUNDOFF LIMITED TO 18 DIGITS PRECISION.
  93 C         IF EITHER IS LARGER THAN U2=0.5/UR, THEN ALL SIGNIFICANCE IS
  94 C         LOST AND IERR=4. IN ORDER TO USE THE INT FUNCTION, ARGUMENTS
  95 C         MUST BE FURTHER RESTRICTED NOT TO EXCEED THE LARGEST MACHINE
  96 C         INTEGER, U3=I1MACH(9). THUS, THE MAGNITUDE OF Z AND FNU+N-1 IS
  97 C         RESTRICTED BY MIN(U2,U3). ON 32 BIT MACHINES, U1,U2, AND U3
  98 C         ARE APPROXIMATELY 2.0E+3, 4.2E+6, 2.1E+9 IN SINGLE PRECISION
  99 C         ARITHMETIC AND 1.3E+8, 1.8E+16, 2.1E+9 IN DOUBLE PRECISION
 100 C         ARITHMETIC RESPECTIVELY. THIS MAKES U2 AND U3 LIMITING IN
 101 C         THEIR RESPECTIVE ARITHMETICS. THIS MEANS THAT ONE CAN EXPECT
 102 C         TO RETAIN, IN THE WORST CASES ON 32 BIT MACHINES, NO DIGITS
 103 C         IN SINGLE AND ONLY 7 DIGITS IN DOUBLE PRECISION ARITHMETIC.
 104 C         SIMILAR CONSIDERATIONS HOLD FOR OTHER MACHINES.
 105 C
 106 C         THE APPROXIMATE RELATIVE ERROR IN THE MAGNITUDE OF A COMPLEX
 107 C         BESSEL FUNCTION CAN BE EXPRESSED BY P*10**S WHERE P=MAX(UNIT
 108 C         ROUNDOFF,1.0E-18) IS THE NOMINAL PRECISION AND 10**S REPRE-
 109 C         SENTS THE INCREASE IN ERROR DUE TO ARGUMENT REDUCTION IN THE
 110 C         ELEMENTARY FUNCTIONS. HERE, S=MAX(1,ABS(LOG10(CABS(Z))),
 111 C         ABS(LOG10(FNU))) APPROXIMATELY (I.E. S=MAX(1,ABS(EXPONENT OF
 112 C         CABS(Z),ABS(EXPONENT OF FNU)) ). HOWEVER, THE PHASE ANGLE MAY
 113 C         HAVE ONLY ABSOLUTE ACCURACY. THIS IS MOST LIKELY TO OCCUR WHEN
 114 C         ONE COMPONENT (IN ABSOLUTE VALUE) IS LARGER THAN THE OTHER BY
 115 C         SEVERAL ORDERS OF MAGNITUDE. IF ONE COMPONENT IS 10**K LARGER
 116 C         THAN THE OTHER, THEN ONE CAN EXPECT ONLY MAX(ABS(LOG10(P))-K,
 117 C         0) SIGNIFICANT DIGITS; OR, STATED ANOTHER WAY, WHEN K EXCEEDS
 118 C         THE EXPONENT OF P, NO SIGNIFICANT DIGITS REMAIN IN THE SMALLER
 119 C         COMPONENT. HOWEVER, THE PHASE ANGLE RETAINS ABSOLUTE ACCURACY
 120 C         BECAUSE, IN COMPLEX ARITHMETIC WITH PRECISION P, THE SMALLER
 121 C         COMPONENT WILL NOT (AS A RULE) DECREASE BELOW P TIMES THE
 122 C         MAGNITUDE OF THE LARGER COMPONENT. IN THESE EXTREME CASES,
 123 C         THE PRINCIPAL PHASE ANGLE IS ON THE ORDER OF +P, -P, PI/2-P,
 124 C         OR -PI/2+P.
 125 C
 126 C***REFERENCES  HANDBOOK OF MATHEMATICAL FUNCTIONS BY M. ABRAMOWITZ
 127 C                 AND I. A. STEGUN, NBS AMS SERIES 55, U.S. DEPT. OF
 128 C                 COMMERCE, 1955.
 129 C
 130 C               COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT
 131 C                 BY D. E. AMOS, SAND83-0083, MAY, 1983.
 132 C
 133 C               COMPUTATION OF BESSEL FUNCTIONS OF COMPLEX ARGUMENT
 134 C                 AND LARGE ORDER BY D. E. AMOS, SAND83-0643, MAY, 1983
 135 C
 136 C               A SUBROUTINE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX
 137 C                 ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, SAND85-
 138 C                 1018, MAY, 1985
 139 C
 140 C               A PORTABLE PACKAGE FOR BESSEL FUNCTIONS OF A COMPLEX
 141 C                 ARGUMENT AND NONNEGATIVE ORDER BY D. E. AMOS, TRANS.
 142 C                 MATH. SOFTWARE, 1986
 143 C
 144 C***ROUTINES CALLED  ZBINU,I1MACH,D1MACH
 145 C***END PROLOGUE  ZBESJ
 146 C
 147 C     COMPLEX CI,CSGN,CY,Z,ZN
 148       DOUBLE PRECISION AA, ALIM, ARG, CII, CSGNI, CSGNR, CYI, CYR, DIG,
 149      * ELIM, FNU, FNUL, HPI, RL, R1M5, STR, TOL, ZI, ZNI, ZNR, ZR,
 150      * D1MACH, BB, FN, AZ, AZABS, ASCLE, RTOL, ATOL, STI
 151       INTEGER I, IERR, INU, INUH, IR, K, KODE, K1, K2, N, NL, NZ, I1MACH
 152       DIMENSION CYR(N), CYI(N)
 153       DATA HPI /1.57079632679489662D0/
 154 C
 155 C***FIRST EXECUTABLE STATEMENT  ZBESJ
 156       IERR = 0
 157       NZ=0
 158       IF (FNU.LT.0.0D0) IERR=1
 159       IF (KODE.LT.1 .OR. KODE.GT.2) IERR=1
 160       IF (N.LT.1) IERR=1
 161       IF (IERR.NE.0) RETURN
 162 C-----------------------------------------------------------------------
 163 C     SET PARAMETERS RELATED TO MACHINE CONSTANTS.
 164 C     TOL IS THE APPROXIMATE UNIT ROUNDOFF LIMITED TO 1.0E-18.
 165 C     ELIM IS THE APPROXIMATE EXPONENTIAL OVER- AND UNDERFLOW LIMIT.
 166 C     EXP(-ELIM).LT.EXP(-ALIM)=EXP(-ELIM)/TOL    AND
 167 C     EXP(ELIM).GT.EXP(ALIM)=EXP(ELIM)*TOL       ARE INTERVALS NEAR
 168 C     UNDERFLOW AND OVERFLOW LIMITS WHERE SCALED ARITHMETIC IS DONE.
 169 C     RL IS THE LOWER BOUNDARY OF THE ASYMPTOTIC EXPANSION FOR LARGE Z.
 170 C     DIG = NUMBER OF BASE 10 DIGITS IN TOL = 10**(-DIG).
 171 C     FNUL IS THE LOWER BOUNDARY OF THE ASYMPTOTIC SERIES FOR LARGE FNU.
 172 C-----------------------------------------------------------------------
 173       TOL = DMAX1(D1MACH(4),1.0D-18)
 174       K1 = I1MACH(15)
 175       K2 = I1MACH(16)
 176       R1M5 = D1MACH(5)
 177       K = MIN0(IABS(K1),IABS(K2))
 178       ELIM = 2.303D0*(DBLE(FLOAT(K))*R1M5-3.0D0)
 179       K1 = I1MACH(14) - 1
 180       AA = R1M5*DBLE(FLOAT(K1))
 181       DIG = DMIN1(AA,18.0D0)
 182       AA = AA*2.303D0
 183       ALIM = ELIM + DMAX1(-AA,-41.45D0)
 184       RL = 1.2D0*DIG + 3.0D0
 185       FNUL = 10.0D0 + 6.0D0*(DIG-3.0D0)
 186 C-----------------------------------------------------------------------
 187 C     TEST FOR PROPER RANGE
 188 C-----------------------------------------------------------------------
 189       AZ = AZABS(ZR,ZI)
 190       FN = FNU+DBLE(FLOAT(N-1))
 191       AA = 0.5D0/TOL
 192       BB=DBLE(FLOAT(I1MACH(9)))*0.5D0
 193       AA = DMIN1(AA,BB)
 194       IF (AZ.GT.AA) GO TO 260
 195       IF (FN.GT.AA) GO TO 260
 196       AA = DSQRT(AA)
 197       IF (AZ.GT.AA) IERR=3
 198       IF (FN.GT.AA) IERR=3
 199 C-----------------------------------------------------------------------
 200 C     CALCULATE CSGN=EXP(FNU*HPI*I) TO MINIMIZE LOSSES OF SIGNIFICANCE
 201 C     WHEN FNU IS LARGE
 202 C-----------------------------------------------------------------------
 203       CII = 1.0D0
 204       INU = INT(SNGL(FNU))
 205       INUH = INU/2
 206       IR = INU - 2*INUH
 207       ARG = (FNU-DBLE(FLOAT(INU-IR)))*HPI
 208       CSGNR = DCOS(ARG)
 209       CSGNI = DSIN(ARG)
 210       IF (MOD(INUH,2).EQ.0) GO TO 40
 211       CSGNR = -CSGNR
 212       CSGNI = -CSGNI
 213    40 CONTINUE
 214 C-----------------------------------------------------------------------
 215 C     ZN IS IN THE RIGHT HALF PLANE
 216 C-----------------------------------------------------------------------
 217       ZNR = ZI
 218       ZNI = -ZR
 219       IF (ZI.GE.0.0D0) GO TO 50
 220       ZNR = -ZNR
 221       ZNI = -ZNI
 222       CSGNI = -CSGNI
 223       CII = -CII
 224    50 CONTINUE
 225       CALL ZBINU(ZNR, ZNI, FNU, KODE, N, CYR, CYI, NZ, RL, FNUL, TOL,
 226      * ELIM, ALIM)
 227       IF (NZ.LT.0) GO TO 130
 228       NL = N - NZ
 229       IF (NL.EQ.0) RETURN
 230       RTOL = 1.0D0/TOL
 231       ASCLE = D1MACH(1)*RTOL*1.0D+3
 232       DO 60 I=1,NL
 233 C       STR = CYR(I)*CSGNR - CYI(I)*CSGNI
 234 C       CYI(I) = CYR(I)*CSGNI + CYI(I)*CSGNR
 235 C       CYR(I) = STR
 236         AA = CYR(I)
 237         BB = CYI(I)
 238         ATOL = 1.0D0
 239         IF (DMAX1(DABS(AA),DABS(BB)).GT.ASCLE) GO TO 55
 240           AA = AA*RTOL
 241           BB = BB*RTOL
 242           ATOL = TOL
 243    55   CONTINUE
 244         STR = AA*CSGNR - BB*CSGNI
 245         STI = AA*CSGNI + BB*CSGNR
 246         CYR(I) = STR*ATOL
 247         CYI(I) = STI*ATOL
 248         STR = -CSGNI*CII
 249         CSGNI = CSGNR*CII
 250         CSGNR = STR
 251    60 CONTINUE
 252       RETURN
 253   130 CONTINUE
 254       IF(NZ.EQ.(-2)) GO TO 140
 255       NZ = 0
 256       IERR = 2
 257       RETURN
 258   140 CONTINUE
 259       NZ=0
 260       IERR=5
 261       RETURN
 262   260 CONTINUE
 263       NZ=0
 264       IERR=4
 265       RETURN
 266       END